Mohajer, Abolfazl; Müller-Stach, Stefan; Zuo, Kang

Special Subvarieties in Mumford-Tate Varieties

Doc. Math. 24, 523-544 (2019)
DOI: 10.25537/dm.2019v24.523-544


Let \(X=\Gamma \backslash D\) be a Mumford-Tate variety, i.e., a quotient of a Mumford-Tate domain \(D=G(\mathcal{R})/V\) by a discrete subgroup \(\Gamma\). Mumford-Tate varieties are generalizations of Shimura varieties. We define the notion of a special subvariety \(Y \subset X\) (of Shimura type), and formulate necessary criteria for \(Y\) to be special. Our method consists in looking at finitely many compactified special curves \(C_i\) in \(Y\), and testing whether the inclusion \(\bigcup_i C_i \subset Y\) satisfies certain properties. One of them is the so-called relative proportionality condition. In this paper, we give a new formulation of this numerical criterion in the case of Mumford-Tate varieties \(X\). In this way, we give necessary and sufficient criteria for a subvariety \(Y\) of \(X\) to be a special subvariety of Shimura type in the sense of the André-Oort conjecture. We discuss in detail the important case where \(X=A_g\), the moduli space of principally polarized abelian varieties.

Mathematics Subject Classification



André-Oort conjecture, period domain, Shimura variety, Higgs bundle


  • 1. S. Abdulali: Conjugates of strongly equivariant maps, Pacific J. Math. 165, 207-216 (1994). DOI 10.2140/pjm.1994.165.207; zbl 0843.14022; MR1300831.
  • 2. F. Andreatta, E. Z.Goren, B. Howard, K. Madapusi Pera: Faltings heights of abelian varieties with complex multiplication, Ann. of Math. 187, Nr. 2, 391-531 (2018). DOI 10.4007/annals.2018.187.2.3; zbl 06841544; MR3744856; arxiv 1508.00178.
  • 3. A. Ash, D. Mumford, M. Rapoport, Y.-S. Tai: Smooth compactifications of locally symmetric varieties, second edition, Cambridge Univ. Press (2010). zbl 1209.14001; MR2590897.
  • 4. J. Carlson, S. Müller-Stach, C. Peters: Period mappings and period domains, Cambridge Studies in Adv. Math, Vol. 168. Cambridge University Press, Cambridge (2017). DOI 10.1017/9781316995846; zbl 1390.14003; MR3727160.
  • 5. P. Deligne: Variétés de Shimura: interprétation modulaire, et techniques de construction de modèles canoniques, Proceedings of Symposia in Pure Mathematics 33, part 2, 247-290 (1979). zbl 0437.14012; MR0546620.
  • 6. B. Edixhoven, A. Yafaev: Subvarieties of Shimura varieties, Ann. of Math.. 157, Nr. 2, 621-645 (2003). DOI 10.4007/annals.2003.157.621; zbl 1053.14023; MR1973057; arxiv math/0105241.
  • 7. M. Green, Ph. Griffiths, M. Kerr: Mumford-Tate groups and domains: Their Geometry and Arithmetic, Annals of Math. Studies 183, Princeton Univ. Press (2012). DOI 10.1515/9781400842735; zbl 1248.14001; MR2918237.
  • 8. Ph. Griffiths, C. Robles, D. Toledo: Quotients of non-classical flag domains are not algebraic, Journal of Alg. Geom. 1(1), 1-13 (2014). DOI 10.14231/AG-2014-001; zbl 1287.53041; MR3234111; arxiv 1303.0252.
  • 9. B. Klingler, A. Yafaev: The André-Oort conjecture, Ann. of Math. 180, Nr. 3, 867-925 (2014). DOI 10.4007/annals.2014.180.3.2; zbl 1377.11073; MR3245009; arxiv 1209.0936.
  • 10. B. Moonen: Models of Shimura varieties in mixed characteristics, in: Galois representations in arithmetic algebraic geometry, LMS Lecture Notes Series Vol. 254, Cambridge University Press, 267-350 (1998). zbl 0962.14017; MR1696489.
  • 11. S. Müller-Stach, E. Viehweg, K. Zuo: Relative Proportionality for subvarieties of moduli spaces of K3 and abelian surfaces, Pure and Applied Mathematics Quarterly 5, Nr. 3, 1161-1199 (2009). DOI 10.4310/PAMQ.2009.v5.n3.a8; zbl 1222.14020; MR1696489.
  • 12. S. Müller-Stach, K. Zuo: A characterization of special subvarieties in orthogonal Shimura varieties, Pure and Applied Mathematics Quarterly 7, Nr. 4, 1599-1630 (2011). DOI 10.4310/PAMQ.2011.v7.n4.a24; zbl 1316.14046; MR2918176; arxiv 1008.5301.
  • 13. J. Pila: O-minimality and the André-Oort conjecture for $\mathbb C^n$, Ann. of Math. 173, Nr. 3, 1779-1840 (2011). DOI 10.4007/annals.2011.173.3.11; zbl 1243.14022; MR2800724.
  • 14. E. Ullmo, A. Yafaev: Galois orbits and equidistribution of special subvarieties: towards the André-Oort conjecture, Ann. of Math. 180, Nr. 3, 823-865 (2014). DOI 10.4007/annals.2014.180.3.1; zbl 1328.11070; MR3245008; arxiv 1209.0934.
  • 15. C. Simpson: Higgs bundles and local systems, Publ. Math. IHES 75, 5-95 (1992). DOI 10.1007/BF02699491; zbl 0814.32003; MR1179076.
  • 16. J. Tsimerman: A proof of the André-Oort conjecture for $A_g$, 2015. arxiv 1506.01466v1.
  • 17. E. Viehweg, K. Zuo: Families with a strictly maximal Higgs field, Asian J. of Math. 7, 575-598 (2003). DOI 10.4310/AJM.2003.v7.n4.a8; zbl 1084.14521; MR2074892.
  • 18. E. Viehweg, K. Zuo: A characterization of certain Shimura curves in the moduli stack of abelian varieties. J. Diff Geom. 66, 233-287 (2004). DOI 10.4310/jdg/1102538611; zbl 1078.11043; MR2106125; arxiv math/0207228.
  • 19. X. Yuan, S.-W. Zhang: On the averaged Colmez conjecture, Ann. Math. 187, Nr. 2, 533-638 (2018). DOI 10.4007/annals.2018.187.2.4; zbl 06841545; MR3744857; arxiv 1507.06903.


Mohajer, Abolfazl
Universit\``{a}t Mainz, Fachbereich 08, Institut f\''ur Mathematik, 55099 Mainz, Germany
Müller-Stach, Stefan
Universit\``{a}t Mainz, Fachbereich 08, Institut f\''ur Mathematik, 55099 Mainz, Germany
Zuo, Kang
Universit\``{a}t Mainz, Fachbereich 08, Institut f\''ur Mathematik, 55099 Mainz, Germany